A370065 Triangle read by rows: T(n,k) is the number of simple graphs on n labeled nodes with k articulation vertices, (0 <= k <= n).

1, 1, 0, 2, 0, 0, 5, 3, 0, 0, 24, 28, 12, 0, 0, 334, 390, 240, 60, 0, 0, 13262, 10776, 6090, 2280, 360, 0, 0, 1106862, 615860, 255570, 92820, 23520, 2520, 0, 0, 175376048, 66625504, 19275424, 5446560, 1429680, 262080, 20160, 0, 0, 52257938968, 13210716600, 2592577512, 520122456, 112145040, 22649760, 3144960, 181440, 0, 0

Offset

0,4

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

S. Selkow, The enumeration of labeled graphs by number of cutpoints, Discr. Math. 185 (1998), 183-191.

Formula

Exponential transform of A370064.

T(n+2, n) = A188588(n + 1).

Example

Triangle begins:
        1;
        1,      0;
        2,      0,      0;
        5,      3,      0,     0;
       24,     28,     12,     0,     0;
      334,    390,    240,    60,     0,    0;
    13262,  10776,   6090,  2280,   360,    0, 0;
  1106862, 615860, 255570, 92820, 23520, 2520, 0, 0;
  ...

Prog

(PARI) \\ Needs G, J defined in A370064.
T(n)={my(v=Vec( ((y-1)*x + serreverse(x/((1-y) + y*exp(G(n)))))/y ), w=Vec(serlaplace(exp(sum(k=1, n, Polrev(J(v[k], k), y)*x^k, O(x*x^n)) )))); vector(#w, n, Vecrev(w[n], n))}
{ my(A=T(8)); for(i=1, #A, print(A[i])) }

Crossrefs

Row sums are A006125.

Column k=0 is A370066.

Cf. A188588, A370064 (connected).

Sequence in context: A143160 A369730 A156387 * A324040 A340116 A361521

Adjacent sequences: A370062 A370063 A370064 * A370066 A370067 A370068

Keyword

nonn,tabl

Author

Andrew Howroyd, Feb 25 2024

Status

approved